I have an outward corner that I need to extend on one side (making a ⊥ from an existing L). What's the lingo for this? I haven't been able to Google it.

On with the actual question: After it's framed up and ready to go, how do you handle the drywall joint at the now-continuous side? It seems like the old corner bead might cause a problem.

 

Yeah, the old corner bead has to be removed. Then tape it just like a butt joint.

 

Because there is often a buildup of joint compound at a corner you might be well advised to remove the drywall away from the corner a stud length or two. or even to stagger the new joints by a stud or two. It will make it easier to taper the butt joints and avoid a hump where the corner was.

You could sand the excess mud off of the corner but it is really hard to sand until you get through the paint. Or cut to the middle of a stud bay back one or two away from the corner and splice the new pieces of rock with a piece of 1/2" plywood about 5" wide fastening half to the existing and leaving half to screw the new piece of rock to. This is described several places here dealing with patches.
This is the way I would do it it is easier to do this than to try to cut to the center of the stud or add a nailer to the side of the stud. There is even a way to put a taper in the but joint if you do it this way. Staple or glue drywall shims to the far outside edge of the plywood then screw closer to the middle when you install the plywood and screw the new piece to it. This will bend the butt ends ever so slightly inward making it easy to get a flat butt joint.